Latoya created a factor tree and wrote the prime factorization of 90 shown below.
A factor tree of 90. 90 branches to 9 and 10. 9 branches to 3 and 3. 10 branches to 2 and 5. The equation is 90 = 2 times 3 times 5.
What is Latoya’s error?
She should not have found the factors of 9.
She should have included an exponent of 2 on the 3.
She should have included 9 and 10 in the prime factorization.
She should have started the tree with 2 times 45.

Step-by-step explanation:

Hi there!

Please see the answer in the picture.

Hope it helps!

1. Approach

One is given a trigonometric equation with and one is asked to prove that it is true. Using the attached image, combined with the knowledge of trigonometry, one can evaluate each trigonometric function. Then one can simplify each ratio to solve. To yield the most accurate result, one has to each of the ratios in a fractional form, rather than simplifying it into a decimal form. Remember the right angle trigonometric ratios, these ratios describe the relationship between the sides and angles in a right triangle. Such ratios are as follows,

[tex]sin(\theta)=\frac{opposite}{hypotenuse}\\\\cos(\theta)=\frac{adjacent}{hypotenuse}\\\\tan(\theta)=\frac{opposite}{adjacent}\\\\csc(\theta)=\frac{hypotenuse}{opposite}\\\\sec(\theta)=\frac{hypotenuse}{adjacent}\\\\cot(\theta)=\frac{adjacent}{opposite}[/tex]

Please note that the terms (opposite) and (adjacent) are relative to the angle uses in the ratio, however the term (hypotenuse) refers to the side opposite the right angle, this side never changes its name. Use these ratios to evaluate the trigonometric functions. Then simplify to prove the identity.

2. Problem (9)

[tex]\frac{sin(60)+cos(30)}{1+sin(30)+cos(60)}=sin(60)[/tex]

As per the attached image, the following statements regarding the value of each ratio can be made:

[tex]sin(60)=\frac{\sqrt{3}}{2}\\\\cos(30)=\frac{\sqrt{3}}{2}\\\\sin(30)=\frac{1}{2}\\\\cos(60)=\frac{1}{2}[/tex]

Substitute,

[tex]\frac{sin(60)+cos(30)}{1+sin(30)+cos(60)}=sin(60)[/tex]

[tex]\frac{\frac{\sqrt{3}}{2}+\frac{\sqrt{3}}{2}}{1+\frac{1}{2}+\frac{1}{2}}=\frac{\sqrt{3}}{2}[/tex]

Simplify,

[tex]\frac{\frac{\sqrt{3}}{2}+\frac{\sqrt{3}}{2}}{1+\frac{1}{2}+\frac{1}{2}}=\frac{\sqrt{3}}{2}[/tex]

[tex]\frac{\frac{2\sqrt{3}}{2}}{1+\frac{1}{2}+\frac{1}{2}}=\frac{\sqrt{3}}{2}[/tex]

[tex]\frac{\frac{2\sqrt{3}}{2}}{1+1}=\frac{\sqrt{3}}{2}[/tex]

[tex]\frac{\sqrt{3}}{1+1}=\frac{\sqrt{3}}{2}[/tex]

[tex]\frac{\sqrt{3}}{2}=\frac{\sqrt{3}}{2}[/tex]

Thus, this equation is true.

2. Problem (10)

Use a similar strategy to evaluate this equation,

[tex]\frac{1-cos(30)}{sin(30)}=\frac{1-cot(60)}{1+cot(60)}[/tex]

Use the attached image to evaluate the ratios.

[tex]cos(30)=\frac{\sqrt{3}}{2}\\\\sin(30)=\frac{1}{2}\\\\cot(60)=\frac{1}{\sqrt{3}}[/tex]

Substitute,

[tex]\frac{1-cos(30)}{sin(30)}=\frac{1-cot(60)}{1+cot(60)}[/tex]

[tex]\frac{1-\frac{\sqrt{3}}{2}}{\frac{1}{2}}=\frac{1-\frac{1}{\sqrt{3}}}{1+\frac{1}{\sqrt{3}}}[/tex]

Simplify,

[tex]\frac{1-\frac{\sqrt{3}}{2}}{\frac{1}{2}}=\frac{1-\frac{1}{\sqrt{3}}}{1+\frac{1}{\sqrt{3}}}[/tex]

[tex]\frac{\frac{2-\sqrt{3}}{2}}{\frac{1}{2}}=\frac{1-\frac{1}{\sqrt{3}}}{1+\frac{1}{\sqrt{3}}}[/tex]

[tex]\frac{\frac{2-\sqrt{3}}{2}}{\frac{1}{2}}=\frac{\frac{\sqrt{3}-1}{\sqrt{3}}}{1+\frac{1}{\sqrt{3}}}[/tex]

[tex]\frac{\frac{2-\sqrt{3}}{2}}{\frac{1}{2}}=\frac{\frac{\sqrt{3}-1}{\sqrt{3}}}{\frac{\sqrt{3}+1}{\sqrt{3}}}[/tex]

[tex]2-\sqrt{3}=\frac{\frac{\sqrt{3}-1}{\sqrt{3}}}{\frac{\sqrt{3}+1}{\sqrt{3}}}[/tex]

[tex]2-\sqrt{3}=\frac{\sqrt{3}-1}{\sqrt{3}}*\frac{\sqrt{3}}{\sqrt{3}+1}}[/tex]

[tex]2-\sqrt{3}=\frac{\sqrt{3}-1}{\sqrt{3}+1}[/tex]

Rationalize the denominator,

[tex]2-\sqrt{3}=\frac{\sqrt{3}-1}{\sqrt{3}+1}[/tex]

[tex]2-\sqrt{3}=\frac{\sqrt{3}-1}{\sqrt{3}+1}*\frac{\sqrt{3}-1}{\sqrt{3}-1}[/tex]

[tex]2-\sqrt{3}=\frac{(\sqrt{3}-1)^2}{3-1}[/tex]

[tex]2-\sqrt{3}=\frac{3-2\sqrt{3}+1}{2}[/tex]

[tex]2-\sqrt{3}=\frac{4-2\sqrt{3}}{2}[/tex]

[tex]2-\sqrt{3}=2-\sqrt{3}[/tex]

Therefore, this equation is also true.

Answer:

4, yes through the middle    5, yes through the middle 6, yes through the middle  all of them reflect from the center

Step-by-step explanation:

[tex]\\ \sf\longmapsto 2x^2+3x^2+3(-1)=5x.x+5x.1[/tex]

[tex]\\ \sf\longmapsto 5x^2-3=5x^2+5x[/tex]

[tex]\\ \sf\longmapsto 5x^2-5x^2-3=5x[/tex]

[tex]\\ \sf\longmapsto 5x=-3[/tex]

[tex]\\ \sf\longmapsto x=\dfrac{-3}{5}[/tex]

Answer:

54

Step-by-step explanation:

162 divided by 3 is 54

Answer:

54 words per minute

Step-by-step explanation:

in 3 minute Rachel types 162 words

so, in 1 minute Rachel types 162words÷ 3 minutes

= 54 words per minute

Hi there!

[tex]\large\boxed{(-3, 8.25)}[/tex]

Solve for the vertex by completing the square:

y = -0.25x² - 1.5x + 6

Factor out a negative:

y = -(0.25x² + 1.5x) + 6

Remember that a square binomial is:

a² + 2ab + b²

We know that:

a² = 0.25, so a = 0.5

1.5 = 2ab, so:

1.5 = 2(0.5)b

b = 1.5

b² = 2.25

Thus:

y = -(0.25x² + 1.5x + 2.25) + 6 - (-2.25)

Simplify:

y = -(0.5x + 1.5)² + 8.25

Find the vertex by factoring out 0.5:

y = -0.5(x + 3)² + 8.25

Thus, the vertex is:

(-3, 8.25)

Answer:

I found x for you, right there.

Answer:  x is 12 and y is 5

Step-by-step explanation:

Answer:

-117 is irrational number

Answer:

Irrational

Step-by-step explanation:

Irrational number can't be written as a faction, -11pie can't be written as a fraction. Therefore it is a irrational number.

Answer:

11 4-passenger cars

Step-by-step explanation:

let 'x' represent the number of 4-passenger cars,then 'x-3' represent the number of 6-passenger cars.

92=4x+6(x-3)

92=4x+6x-18

92=10x-18

92+18=10x

110=10x

x=110/10

x=11

Therefore there are 11 4-passenger cars

Answer:

{1, 0, 16}

Step-by-step explanation:

given..

j(x) = x^2-2x+1

put all given values of domain (1,0 and 5 ) in the equation..

the values you get are range of the function

Answer: 1

Step-by-step explanation

0(2)-2(0)+1=0+0+1= 1

1(1)-(2)(1)+1=1-2+1= 1

(5)(2)-(2)(5)+1=10-10+1= 1

Given:

The equation for the linearized regression line is:

[tex]\log y=0.30x+0.296[/tex]

where x represents number of weeks and y be the customer's stock.

To find:

The number of weeks that will pass before the value of the stock reaches $200.

Solution:

We have,

[tex]\log y=0.30x+0.296[/tex]

Substituting y=200, we get

[tex]\log (200)=0.30x+0.296[/tex]

[tex]2.301=0.30x+0.296[/tex]

[tex]2.301-0.296=0.30x[/tex]

[tex]2.005=0.30x[/tex]

Divide both sides by 0.30.

[tex]\dfrac{2.005}{0.30}=x[/tex]

[tex]6.6833333=x[/tex]

[tex]x\approx 6.7[/tex]

Therefore, the correct option is C.

Answer:

5, 2

Step-by-step explanation:

hope u got it.........

Answer:

y = 5x + 2

Step-by-step explanation:

The equation of a line in slope- intercept form is

y = mx + c ( m is the slope and c the y- intercept )

Given

- 5x + y = 2 ( add 5x to both sides )

y = 5x + 2 ← in slope- intercept form

Answer:

No, they cannot. Some polynomials are different than others and can be factored in different ways. Some can't even be factored at all.

Perfect Square Trinomials have a pattern with factoring and can be written as one binomial squared.

Difference of Squares has a pattern of (x - y)(x + y).

Difference/Sum of Cubes have formulas for their patterns.

Prime Polynomials simply cannot be factored.

Step-by-step explanation:

Brainliest, please! (Almost an Ace)

Answer:

11/12

Step-by-step explanation:

2/3 + 1/4

Get a common denominator of 12

2/3 *4/4   + 1/4 *3/3

8/12 + 3/12

11/12